Simplify arccsc( square root of 2)

Problem

arccsc(√(,2))

Solution

1. Identify the definition of the inverse cosecant function. The expression y=arccsc(√(,2)) is equivalent to finding an angle y such that csc(y)=√(,2) within the range [−π/2,0)∪(0,π/2]

2. Convert the cosecant equation into a sine equation using the reciprocal identity csc(y)=1/sin(y)

sin(y)=1/√(,2)

3. Rationalize the denominator of the fraction to recognize a standard trigonometric value.

1/√(,2)=√(,2)/2

4. Determine the angle y in the first quadrant that satisfies the equation sin(y)=√(,2)/2

y=π/4

Final Answer

arccsc(√(,2))=π/4

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